====== Notes ======

===== In Progress =====

  * [[research:draft:2011:relaxed_assumptions:start|]] --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-10-08 11:53//
===== Done =====

  * finished [[research:draft:2011:continuous_types]]
  * finished [[research:draft:2011:implicit_competition:start|]]
  * showed that "asymmetric" process (with a focal individual) isn't a problem, equivalent to assumption that rate constants of two processes, one with each kind of partner, are the same. --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-07-29 20:34// 
  * studied "asymptotic expansion" in [[http://arxiv.org/abs/1108.1999]].  Looks like just another way to write what I'm doing -- I don't see any advantages. --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-08-26 23:57//
  * nondimensionalized [[research:draft:2011:implicit_competition:start|]] and showed the dynamics can be characterized by just the Allee strength parameter, \(A\). --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-09-09 23:19//
  * chose the best sign in [[research:draft:2011:implicit_competition:start|]] for the radical solutions of \(r,K,n_A\) in terms of \(u_0,u_1,u_2,...\) -- define \(K\equiv (u_1+R)/2 u_2\) (ie. //positive// radical) where \(R\equiv\sqrt{u_1^2+4 u_0 u_2}\).  Then \(r,K>0\) and \(K\) stable for all sustainable((biologically meaningful)) cases.  --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-09-17 00:09//

===== To Do =====

  * regroup.  What is the big picture here?  Where am I in the process? --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-09-09 23:56//
  * trying to get meaningful r, K, nA, etc directly from rate constants is going nowhere.  Want connection so you can make testable predictions.  What about going in reverse?  Eg. derive rate constants from population parameters and test, for example, what the average spontaneous death rate is (can be compared to average lifetime in the absence of competition). --- //[[rik.blok@ubc.ca|Rik Blok]] 2011-10-08 11:50//
==== Stochastics ====

[[~rikblok]] Thu Apr  8, 2010 @  9:49PM

The most important contribution from this paper is the (weak) Allee effect.  It implies that a depressed population will stay depressed longer than expected from the logistic equation.  This becomes more important when we consider demographic stochasticity.  Then there will probably be a significantly increased risk of extinction because of the extended low.  Two ways to investigate: (1) run out many replicates and collect extinction statistics (vs logistic).  Or (2) build master equation and derive extinction probability.

==== Data sets ====

[[~rikblok]] Thu Apr  8, 2010 @  9:49PM

Lab suggestion: It seems the logistic equation is used excessively.  We're arguing that in sexual populations there should be a small-population correction.  If that's right we should be able to find data sets that support this.  Plot dn/dt versus n and see if it is quadratic (logistic) or if there's a depression (Allee effect) for small n.  To be really convincing, this depression should be common in sexual populations and rare in asexuals.

Reply: I think [[#courchamp08 | [courchamp08] ]] covers this better than I could.  Maybe I should stick to the mechanistic explanation for processes that allow for approximate logistic dynamics (with an Allee effect) in sexual populations.  In all the cases I've thought of, the Allee effect emerges from "mate finding" problems at low population densities.

===== Title =====

I was thinking: "Mating Logistics" because it's punny.  (Plus some subtitle.)

===== Reasoning =====

  - Logistic equation is asexual
  - Adding sex via underlying reaction-kinetics generally breaks logistic form
  - Can only recover logistic in special cases (eg. if a solo parent is involved in a rate-limiting step)
  - Deviations from logistic tend to occur as Allee effects (where there's a depressed growth rate for very low population densities)
  - Results are fairly general, found for many ways of writing mating processes
  - Allee effect means populations are less robust against decimation than would be expected from logistic.  May be important in conservation ecology

===== Layout =====

Here's how I think the paper should be laid out:

  - Introduction - historical introduction to logistic equation, problems, recovery of logistic w/ sex, Allee effect
  - The logistics of sex - various examples of derivation (estrous, gestation, monogamy, etc)
  - General solution - demonstrate logistic dynamics & Allee effect, predictions
  - Summary - repeat

===== Introduction =====

Here's a start at the intro:

Malthus introduced the concept of geometric population explosion with the equation x'=x.
"Population, when unchecked, increases in a geometrical ratio."
"In no state that we have yet known has the power of population been left to exert itself with perfect freedom."
Malthus understood this couldn't go unchecked and Verhulst (1845) introduced an "interference" term that had minimal effect when the population was small but slowed growth of large populations.
Verhulst's logistic equation very popular in ecological literature. [Reference feller40.]

Maynard Smith wrote "The logistic equation is best regarded as a purely descriptive equation". [Reference Models in Ecology, 1978.]
Nevertheless, it became ubiquitous in describing population growth.
Kingsland (1982) argues that it's rise to dominance was not a product of good science but a result of sociological forces.

One common objection to the overuse of the logistic model is that it is correlational, the curve is fit to the population, rather than explanatory. [Again refer to feller40.]
But logistic equation can be a prescriptive model.  We can in fact derive logistic (Verhulst) equation for population dynamics from an understanding of the processes that occur at level of individuals.
Assumes large, well-mixed population (so demographic stochasticity can be neglected).
Method (reaction kinetics) clearly exposes assumptions about which of the processes involved are most important.
For Verhulst's logistic equation we require asexual reproduction and one-on-one competition.
N to 2N, 2N to N.
(Another form of competition--limited resources, such as nesting sites--also produces logistic dynamics.)
Note common ecological terms, r and K, are related to biological processes.  With death process, r = b - d, K = .

Logistic equation is therefore a desription of an asexual population.
But has been often applied to sexual populations.
[Report some statistics on use of "logistic equation" and "sexual reproduction" in publications.]
If we want a prescriptive model of sexual population dynamics we will have to derive it from first principles.  
Reaction kinetics provides a method to do this.

Adding sex in the most straightforward way "breaks" the model, leading to unstable dynamics (population explosion or extinction, depending on parameter combinations).
2N to 3N, 2N to N [Derive dynamics.]

Here we explore reaction kinetics processes that explicitly include sex but can produce roughly logistic dynamics.
Dynamics approach logistic when intrinsic growth rate r is tiny.
Will show there needs to be a rate-limiting process involving a solo reactant.
Rate limiting step produces slow intrinsic growth.
We find that deviations from logistic (when r bigger (or process no longer rate-limiting?)) tend to occur as weak Allee effects (there's a depressed per capita growth rate for very low population densities).  

Results are fairly general, found for many ways of writing mating processes (eg. Male + Female --> ..., 2 Unmated --> Mated + Unmated, Anestrous --> Estrous, 2 Unmated --> Couple, etc).

We predict that certain sexual populations that appear to obey logistic dynamics will show Allee effects when the populations are small.

===== Aside: Phenomonology =====

I've noticed a number of papers make the mistake of treating the [[http://en.wikipedia.org/wiki/Phenomenology_%28science%29 | phenomenological]] (descriptive) logistic equation as explanatory.  In "Evolutionary Consequences of Basic Growth Equations" (1992) Ginzburg even tried to add a mechanism to the equation.  In effect he took a description of nature, misinterpreted it as an explanation, and added another mechanism to try explain something else.  That's analogous to saying "Here's a pretty flower--let's add an eyeball to it."  The problem is it no longer describes //or// explains anything natural.  It appears this type of mistake is common and has been repeatedly admonished but still occurs.  Could this paper be a suitable medium to reiterate it again?

===== Applications =====

At my last lab meeting it was suggested that this research would be more valuable if concrete examples could be used.  We brainstormed a few ideas.  Florence had a good suggestion:

On 2011-06-23 4:54 PM, Florence Debarre wrote:
> Hi,
> 
> Another example with the 2-type particle system could be a S I1 I2 
> epidemiological model, with fixed total host densities (s + i1 + i2 = 
> constant, so that S is like "available space"), with parasite mutations 
> and superinfections. In the simple version without mutations and 
> superinfections, you recover the logistic equation (see e.g. 
> http://www.normalesup.org/~fdebarre/papers/Debarre-Lenormand-Gandon_2009_PLoS-Comp-Biol.pdf ) 
> but you might get Allee effects if one parasite is a very good 
> within-host competitor, but a poorer "infector". And here you can even 
> have asymmetry competition (i.e. rates which depend on who is the 
> focal), because the outcome of I1 + I2 depends on who is attacking the 
> other.
> 
> Hope it helps,
> 
> Florence
> 

===== Problems =====

===== References =====

[[http://arxiv.org/abs/0907.0989 | A population facing climate change: joint influences of Allee effects and environmental boundary geometry]] - emphasizes the practical significance of an Allee correction to logistic growth

[[#courchamp08]][[#courchamp08 | [courchamp08] ]] [[http://books.google.ca/books?id=ILPEdtKc0roC&lpg=PP1&ots=MrdqbUOzFF&dq=courchamp%20Allee%20Effects%20in%20Ecology%20and%20Conservation%202008&pg=PA257#v=onepage&q&f=false | Allee effects in ecology and conservation]]  By Franck Courchamp, Luděk Berec, Joanna Gascoigne (2008) - Table 2.1 lists many causes of Allee effects (including mate finding) with real-world examples and references; Section 3.3 (p.96-7) discusses how to detect an Allee effect, with references

[[http://arxiv.org/abs/1008.4854 | Extinction rates of established spatial populations]] - discusses Allee effect on extinction